# Download e-book for kindle: A guided tour of mathematical physics by Snieder R.

By Snieder R.

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12) for the electric eld and Gauss's law to show that within the hollow sphere the electric eld vanishes: E(r) = 0 for r < R. This result implies that when a charge is placed within such a spherical shell the electrical eld generated by the charge on the shell exerts no net force on this charge the charge will not move. Since the electrical potential satis es E = V , the result derived in problem d implies that the potential is constant within the sphere. 12) (which implies that the eld of point charge decays as 1=r2 ).

6) R3 ^r : Plot the gravitational eld as a function from r when the distance increases from zero to a distance larger than the radius R. Verify explicitly that the gravitational eld is continuous at the radius R of the sphere. 17) for the gravitational eld. As an example we will consider a hollow spherical shell with radius R. On the spherical shell electrical charge is distributed with a constant charge density: = const. 12) for the electric eld and Gauss's law to show that within the hollow sphere the electric eld vanishes: E(r) = 0 for r < R.

5) we computed the magnetic eld generated by a current in a straight in nite wire. The eld equation r B = 0J (5:12) again was used to compute the eld away from the wire. 13) contained an unknown constant A. 12) was only used outside the wire, where J = 0. 5) therefore did not provide us with the relation between the eld B and its source J. The only way to obtain this relation is to integrate the eld equation. This implies we have to compute the integral of the curl of a vector eld. The theorem of Stokes tells us how to do this.